Consider two ceiling fans with the same properties: drag coefficient, power input, total mass, etc. The only difference is F1 has a greater moment of inertia than F2.

Assume the airstream from one fan does not affect the rotation of the other fan at all.

Start both fans at the same time, with the same amount of power input. After a very long time, are both fans rotating at the same speed? Or is F1 rotating faster than F2? Or is F2 rotating faster than F1?


1 Answer 1


They will eventually reach the same speed.

When they reach terminal velocity the power erogated will be spent to counterbalance the drag. Since $P=T\omega=(\eta\omega)\omega=\eta\omega^2, $ where $T=\eta\omega$ is the torque of the drag, if $P$ and $\eta$ are equal between the two fans also $\omega$ will be (this is still true also if $T=\eta\omega^2$ or any other power).

If you are not convinced by this reasoning the explicit equation of motion is (assuming the power $P$ delivered is constant) $$\omega(t)^2=\frac{P}{\eta}\left(1-\exp\left(-\frac{2\eta}{I}t\right)\right)$$ and it is clear that the terminal velocity is again $P/\eta$ (you can verify this by substituting into the differential equation $I\dot{\omega}=T_\text{ext}-T_\text{drag}=T_\text{ext}-\eta\omega$ after multiplying both sided by $\omega$ to get $I\dot{\omega}\omega=P-\eta\omega^2$).

  • $\begingroup$ but the torque of drag depends on the angular velocity of the fan. And the angular velocity of F1 is initially smaller than F2. Ia there some way you can show that steady state is reached after a long time? $\endgroup$ Apr 23, 2017 at 15:16
  • $\begingroup$ Sorry, I misread your question. I have edited my answer. $\endgroup$
    – pp.ch.te
    Apr 23, 2017 at 15:29

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